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Closure & Closed Sets in metric space

  1. Feb 25, 2010 #1
    Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F.

    Claim 1: F is contained in the clousre of F.
    Claim 2: The closure of F is closed.


    How can we prove these formally?

    For claim 1, I think we have to show x E F => x E cl(F).
    For claim 2, we need to prove that cl(F) contains all limits of sequences in cl(F).
    But how can we prove these?

    I know these are supposed to be basic facts, but my book never gives examples of how to prove these from the definitions given above...and I have no clue...
    Any help is appreciated!
     
  2. jcsd
  3. Feb 25, 2010 #2
    For claim 1, yes, you want to show that if x is in F, then x is in the closure of F. For some element x in F, can you find a sequence in F that converges to x?

    For claim 2, simply apply the definition of closure to what is being asked
     
  4. Feb 25, 2010 #3
    1) So we have to show that if x E F, then x is the limit of a sequence in F??
    How can we show it? I can't figure it out...

    2) The closure of F is the set of all limits of sequences in F.
    But we need to prove that cl(F) contains all limits of sequences in cl(F). How?

    Can somebody explain a little more, please?
    Thanks!
     
  5. Feb 25, 2010 #4
    (1) Think about constant sequences.

    (2) Clearly cl(F) = F u F' where F' is the set of limit points of F. Use this and think about how 'adding' F' to F changes the sequences in F - all you are doing is 'completing' the sequences in F, you are not introducing anything new, so there would be no more potential limit points to consider.
     
  6. Feb 26, 2010 #5
    (2) Let {c_n} be a convergent sequence in cl(F). We want to show that it has its limit in cl(F). We can find a sequence {a_n} of elements in F, such that for every n, d(a_n,c_n)<1/n (because c_n is the limit of some sequence in F). Therefore lim c_n = lim a_n in cl(F).
     
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